Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Traces of Intertwining Operators for the Yangian Double

The traces over infinite-dimensional representations of the central extended Yangian double for the product of operators which intertwine these representations are calculated. For the special combinations of the intertwining operators, the traces are identified with form factors of local operators in the SU(2)-invariant Thirring model. This identification is based on the identities which are deformed analogs of the Gauss–Manin connection identities for the hyperelliptic curves.     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.     

Logarithmic Intertwining Operators and the Space of Conformal Blocks Over the Projective Line

We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This can be viewed as a generalization of Zhu??s result for ordinary intertwining operators among ordinary modules.     

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.     

On the Two Spectra Inverse Problem for Semi-infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg–Marchenko theorem for Schrödinger operators on the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.     

q类似玻色算符的逆算符及其应用

在q类似Fock空间中引入了q类似玻色算符的逆算利用的性质,构造了增光子相干态的q形变形式并讨论其完备性关系.     

Generalization of Susskind-Glogower Phase Operators and Inverse Field Operators to q-Deformed Case

We analyze some properties of Susskind-Glogower (SG) phase operators exp(iΦ) and exp(-iΦ) by making use of inverse field operators. The generalization of the analysis to q-deformed case is given.     

On One Inverse Problem of Phase Transformation in Solids

The process of diffusional phase transformation described by the two-phase Stefan-type model with the free boundary has been considered. In terms of this model, an inverse problem of boundary conditions that provide the displacement of the free boundary by a given law has been stated. The problem has been solved numerically using a computational algorithm that exploits the front rectification method and a variational approach with local regularization.     

Inverse of Radiation Field Operators and the Generalized Jaynes-Cummings Model

We generalize the standard Jaynes-Cummings model (JCM) to a model Hamiltonian with the radiation field operators being the inverse of a harmonic oscillator's creation and annihilation operators. Some new commutative relations about the inverse operators are derived and the generalized JCM Hamiltonian's eigenstates are derived.     

On the Problem of Emergence of Classical Space—Time: The Quantum-Mechanical Approach

The Riemannian manifold structure of the classical (i.e., Einsteinian) space-time is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The space H is associated with the space of states of a macroscopic test-particle in the universe. The spatial localization of state of the particle through its interaction with the environment is associated with the selection of a submanifold M of realization H. The submanifold M is then identified with the classical space (i.e., a space–like hypersurface in space-time). The mathematical formalism is developed which allows recovering of the usual Riemannian geometry on the classical space and, more generally, on space and time from the Hilbert structure on S. The specific functional realizations of S are capable of generating spacetimes of different geometry and topology. Variation of the length-type action functional on S is shown to produce both the equation of geodesics on M for macroscopic particles and the Schrödinger equation for microscopic particles.     

Semi-Groups and Time Operators for Quantum Unstable Systems

We use spectral projections of time operator in the Liouville space for simple quantum scattering systems in order to define a space of unstable particle states evolving under a contractive semi-group. This space includes purely exponentially decaying states that correspond to complex eigenvalues of this semi-group. The construction provides a probabilistic interpretation of the resonant states characterized in terms of the Hardy class.     

Adaptive Cluster Expansion for the Inverse Ising Problem: Convergence,Algorithm and Tests

We present a procedure to solve the inverse Ising problem, that is, to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of spins, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.     

Inverse Problem for Polynomials Orthogonal on the Unit Circle

Polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated. It is shown that the coefficients are in fact quasi-periodic. The measures associated with these quasi-periodic coefficients are exhibited and necessary and sufficient conditions relating quasi-periodicity and spectral measures of this type are given. Analogs for polynomials orthogonal on subsets of the real line are also presented. Received: 21 May 1997 / Accepted: 28 July 1997     

Positive-Operator-Valued Measures and Projection-Valued Measures of Noncommutative Time Operators

Some relationships between two differentconcepts of noncommutative time operators are discussed.One is the concept of a Hermitian, but not self-adjointtime operator T_B based on apositive-operator-valued measure for a dynamical observable B. The otheris the concept of a self-adjoint time operatorT_L obtained in the Liouville representation,a special case of the standard representation of quantumtheory. Conditions are indicated under which aself-adjoint extension of T_B leading toT_L can be constructed. Similarities with thenotions of consistent and inconsistent histories areindicated. Conceptual issues as to the interpretation of the different timeoperators are outlined with particular emphasis on thenotion of temporal nonlocality.     

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.